Sets with more differences than sums

We show that a random set of integers with density 0 has almost always more differences than sums. This proves a conjecture by Martin and O'Bryant

A pp-group with positive Rank Gradient

We construct for $d\geq 2$ and $\epsilon>0$ a $d$-generated $p$-group $\Gamma$, which in an asymptotic sense behaves almost like a $d$-generated free pro-$p$-group. We show that a subgroup of index $p^n$ needs $(d-\epsilon)p^n$ generators, and that the subgroup growth of $\Gamma$ satisfies $s_{p^n}(\Gamma)>s_{p^n}(F_d^p)^{1-\epsilon}$, where $F_d^p$ is the $d$-generated free pro-$p$-group. To do so we introduce a new invariant for finitely generated groups and study some of its basic properties

Partitions which are p- and q-core

Let p and q be distinct primes, n an integer with n > p2q2. Then there is no partition of n which is at the same time p- and q-core. Hence there is no irreducible representation of Sn which is of p- and q-defect zero at the same time

The irrationality of some number theoretical series

We prove the irrationality of some factorial series. To do so we combine methods from elementary and analytic number theory with methods from the theory of uniform distribution